Anomalous X-Ray Scattering

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Abstract
Unlike gases, liquids and glasses are condensed matter, with atomic densities almost equal to those in crystalline solids. Each atom interacts strongly with its neighbors, and tries to set up an environment most comfortable to that atom. This creates "short-range order (SRO)" which is both physical and chemical. In other words where the neighbors are and what neighbor you have. The total scattering tells you mostly the physical SRO, not the chemical, or compositional, SRO (CSRO), except that the bond lengths are so distinct that each peak in the pair-distribution function (PDF) can be assigned chemical identity. But such cases are exceptions than the norm. Often peaks with different chemical identities overlap and cannot be resolved. Anomalous x-ray scattering (AXS) is the technique of choice to resolve CSRO with x- rays. It takes advantage of the strong energy dependence of the atomic scattering factor near the absorption edge. The x-ray scattering factor of an atom is energy independent, most of the time. But when the incident energy of x-ray is equal to the absorption edge, say of the K-shell, the incident x-ray photon resonates with the 1s-2p excitation, and induces emission which is out-of phase. The consequence is the reduced scattering factor, in the close vicinity of the absorption edge. For this reason this technique is also called the resonant elastic x-ray scattering (REXS). In a multi-component system, since the x-ray scattering factor of other elements do not change much in a narrow energy range near the absorption edge of one element, measuring the scattering intensities at two incident energies just below the absorption edge of one element isolates the structure factor involving that element, and yields the differential PDF (DDF) (1) as shown below. AXS is also a powerful technique when combined with small angle x-ray scattering (SAXS). SAXS is sensitive to mesoscopic density fluctuation. When combined with AXS (SAAXS) it can probe the mesoscopic compositional fluctuation. Whereas the theory of AXS appears straightforward, the practice of AXS is not. First, the energy of the absorption edge is fixed by atomic physics. The energy of the incident x-ray, Ei, is directly related to the scattering vector, Q, by Q = 2ksinθ = (2Ei/c)sinθ, where θ is the diffraction angle. Since sinθ ≤ 1, Q ≤ 2Ei/c; the energy of the absorption edge limits the range of Q. For light elements it is too low to gain sufficient information, except for SAAXS. The second reason is the so-called x-ray Raman effect, induced electronic transition below the absorption edge. If the incident energy is above the absorption edge the sample strongly absorbs x-ray and emits fluorescence, emission of the characteristic x-ray from the sample, which interferes with the measurement. Thus we have to choose the incident energy below the absorption edge. But even below the edge fluorescence occurs because of the x-ray Raman effect. Therefore the AXS setup has to be made to exclude the fluorescent radiation. This, in principle, can be done by using an analyzer crystal. Now, most of the x-ray diffraction measurements are done to determine the position of the Bragg peaks in Q-space with high accuracy, and the accuracy of the intensity measurement is secondary in importance. Thus the analyzer crystal is customarily used in high-resolution diffraction measurement. However, in